Page - 100 - in Contributions to GRACE Gravity Field Recovery - Improvements in Dynamic Orbit Integration, Stochastic Modelling of the Antenna Offset Correction, and Co-Estimation of Satellite Orientations

tracking observable (see eq. (6.4.58)) can be condensed to ∆lsst= ρ˙sst+∆ρ˙AOC+e (8.1.1) with ρ˙sst the reduced range rate ρ˙sst= ρ˙KBR+∆ρ˙TOF+∆ρ˙Iono− ρ˙COM . (8.1.2) Before, only one stochastic model was estimated for the ll-SST observations under the assumption of stationarity. A priori knowledge of the spacecraft orientation uncertain- ties can be exploited to separate the noise due to the non-stationary AOC process from the remaining noise. The remaining noise sources will continue to be treated under the assumption of stationarity. In effect, the stochastic model will be ˆ˙ρsst= ρ˙sst+ eˆsst , eˆsst∼N(0,Σsst) (8.1.3) ∆ ˆ˙ρAOC=∆ρ˙AOC+ eˆAOC , eˆAOC∼N(0,Σ∆ρ˙AOC) (8.1.4) with Σsst determined through VCE as discussed in section 6.5, and Σ∆ρ˙AOC to be discussed in the following sections. The complete stochastic model for the reduced observations∆lsst is obtained through variance propagation as Σll=Σsst+Σ∆ρ˙AOC . (8.1.5) 8.1.1 Uncertainties in Computing the Antenna Offset Correction As discussed in section 6.2, the adapted ITSG-Grace2016 sensor fusion algorithm gives complete variance-covariance matrices for the orientation of the GRACE satellites. Let these be ΣˆAαˆαˆ for GRACE-A and Σˆ B αˆαˆ for GRACE-B. These covariances represent the uncertainty of the small angle rotation from the nominal orientation of the satellite to the observed instantaneous orientation. This is the rotation R˜α from eq. (6.2.4). For one epoch, the AOC as given in eq. (4.3.6) can be approximated through ∆ρAOC= 〈RAcA,e〉+〈RBcB,e〉 , (8.1.6) whereRs, s∈A,B is the observed rotation from the SRF to the CRF, R˜SRFCRF, for that satellite. The APC vectors are given in the SRF, the satellite baseline e in the CRF. Rewriting eq. (8.1.6), the AOC can also be expressed in terms of the three-dimensional opening angle βbetween the APC vector and the satellite baseline: ∆ρAOC=‖cA‖·cosβA+‖cB‖cos ·βB , βs=arccos 〈 1 ‖cs‖Rscs,e 〉 . (8.1.7) The opening angle for GRACE-A is visualized in fig. 8.1. This formulation is useful to keep in mind as, due to the small values ofβ, the opening angle can often be used as a proxy for the magnitude of the AOC that is intuitive to conceptualize and classify. Chapter8 Star Camera Observations and Uncertainties100